### 6.780

### Semiconductor Manufacturing

### Spring 2003

**Problem Set #2 **

Issued: Wednesday, Feb. 12, 2003

Due: Friday, Feb. 21, 2003

Please write and hand in solutions to the following problems or scenarios. Write your solutions from the perspective of an engineer sending an explanation to a coworker (or a customer, or a supplier) of your solution. That is to say, you must write supporting text explaining equations and formulas that you use; a few mathematical expressions and a number or plot at the end will not earn full credit.

**Problem 1 **

We are considering the recalibration of a low-pressure chemical vapor deposition (LPCVD) furnace. The furnace will be shut down for repairs if a significant difference is found between the thermocouple read ings that are controlling the deposition temperature at the two ends of the tube. The following tempera tures have been measured during a test run:

**Table 1: Measured Left/Right Temperatures in LPCVD Test Run (°C) **

Left 606.5 605.0 605.5 605.5 606.2 606.5 603.7 607.7 607.7 604.2

Right 604.0 604.5 605.5 605.7 605.5 605.2 606.0 606.5 607.7 604.2

**Part a. **Do these temperatures have the same mean on both ends of the tube? (

α

= 0.05)

**Part b. **Find the 90% confidence interval of the ratio of the two variances (Left/Right).

**Part c. **Find the sample size needed to confirm a 1°C deviation with a

β

of 10% when the

σ

is known to be 1°C (assuming

α

= 0.05 as in part a).

**Problem 2 **

A normally distributed quality characteristic is monitored by a control chart with 3-

σ

control limits.

**Part a. **Develop a general expression for the probability that a point will plot outside the control limits when the process is really in control.

**Part b. **Develop a general expression for the probability that 2 out of 3 consecutive points will lie on the same side of the center line, but outside the 2-

σ

line, when the process is really in control (this is the

WECO #2 rule).